How do I find an integer value for which an expression is non-prime?

Solution 1:

How about $n=41$?

In general, if you choose $n$ so that all of the terms in a sum are divisible by the same number, then the whole sum will be divisible by that number.

Edit: My understanding is that your approach was to set $n^2-n+41 = 2n+1$ and look for integer solutions. But this is quite a strong condition: you're saying not just that $n^2-n+41$ is odd, but that it's the particular odd number $2n+1$. This is a quadratic in $n$, so it has at most $2$ solutions - it's not particularly surprising that it doesn't have integer solutions.

But for any integer value of $n$, $n^2-n+41$ is odd. If you just want to express that $n^2-n+41$ is odd, the relevant equation is $n^2 -n +41 = 2k+1$. This equation has exactly one integer solution for every value of $n$: an example is $n = 41$, $k = 840$.

Solution 2:

Though $n = 41$ is an obvious solution, your question 1 (b) is really to be able to find for which $n$, the given expression is not prime. (Though it asks 'an integer', we can try to make a generalization to find all integers).

One approach would be to write it as $n^2 - (n - 41)$. If we make $(n - 41)$ a square, then obviously the given expression factorizes.

So $\color{blue}{n = 41, 42, 45, 50, 57}$ and so on are values of $n$ for which the given expression is not prime. I think Alex has answered the rest.